Skill: systems-inequalities

Systems of Inequalities

Inequalities shade regions, not lines. Graph each one — solid or dashed, above or below — and the solution is where the shadings overlap. Learn to test points, find vertices, and read feasible regions off a graph.

1 Graph one inequality first

Every system starts with single inequalities. Two decisions: which side to shade, and whether the boundary is solid or dashed.

Two rules

y > mx + b → shade ABOVE · y < mx + b → shade BELOW

≤ or ≥ → SOLID boundary (points on it count) · < or > → DASHED boundary

Shade toward the inequality sign; solid boundary includes the line, dashed excludes it.

Test-point trick: unsure which side? Plug in (0, 0) (if it is not on the line). If it satisfies the inequality, shade the side containing the origin.

2 Worked examples

From single inequalities to full systems.

Example 1 Test a point

Does (1, 4) satisfy y ≥ 2x + 1?

2(1) + 1 = 3, and 4 ≥ 3. Yes.

Example 2 Which side?

For y < −x + 3, shade below the line.

Example 3 Solid or dashed?

y > 3x − 2 has a strict sign, so the boundary is dashed.

Example 4 A system

Solve y ≥ x − 1 and y ≤ −x + 3.

Boundaries cross where x − 1 = −x + 3, i.e. at (2, 1) — the vertex of the feasible region. The solution is the overlap of “above line 1” and “below line 2”, a wedge opening upward.

3 Common mistakes

Three traps, each with the wrong version and the fix.

1. Shading the wrong side

Wrong: shading below for y > 2x + 1. Right: the sign points the way — > shades above.

2. Solid vs dashed swapped

Wrong: dashed line for y ≤ x. Right: ≤/≥ include the boundary → solid.

3. Forgetting the overlap

Wrong: shading both inequalities but circling one side only. Right: the solution of a system is where the shadings overlap.

4 Quick checks

Try these yourself, then reveal the answer.

Does (0, 5) satisfy y ≥ 2x + 1?

2(0) + 1 = 1, and 5 ≥ 1: yes.

y > −2x + 4: above or below? Solid or dashed?

Above the line, dashed boundary.

Where do the boundaries y = x and y = −x + 4 cross?

x = −x + 4 gives x = 2: (2, 2) — the vertex.

5 Key points

Remember

y > / ≥ → shade above; y < / ≤ → shade below.
≤ and ≥ → solid boundary; < and > → dashed.
Test point (0, 0) decides the side when you are unsure.
A system’s solution is the overlap of all shadings.
Vertices sit where two boundary lines cross — they satisfy every inequality.

Key vocabulary

Feasible region

The overlapping shaded region satisfying every inequality in the system.

Boundary line

The line y = mx + b that edges an inequality’s shading.

Vertex

A corner of the feasible region where two boundaries cross.

Strict inequality

Uses < or >; the boundary is dashed and excluded.

Practice shading Open the region grapher Explore in the lab